Just as you can apply one function to the results of another, you can use the conclusion of one argument as the premise for another. However, this chain of logical arguments has to have a beginning--a starting premise that is not itself the conclusion of any other argument. By definition, this premise is illogical. Thus, we can say not only that everyone is illogical, but that illogic is the necessary foundation for logic.

When engaging in philosophical discussions of the type that are popular among some people on Everything, it is useful to apply logic to each person's arguments. If their logic holds up, you can fairly quickly come back to the illogical judgments that serve as someone's premises. Of course, these premises are often rooted so deeply in someone's nature that they may be uncomfortable about even discussing them, much less having them challenged. However, since these deep premises are the roots of our beliefs and ideas, it is absolutely vital that they be found and examined.

To Yam: Assertions can still be defined as "logical" or "illogical," although that doesn't necessarily say a lot about their truthfulness.

And I agree that assertions should be examined for the evidence in their favor, but what criteria do we apply to determine what constitutes evidence? The variations in rules of evidence are themselves axioms that are usually not based in rationality but in a commitment to a particular world-view.

Thus, no assertion can be strictly defined as logical, which dilutes the designation "illogical" somewhat. The key is that logic is not an absolute state, but a process. "Logical" does not mean true - it means in keeping with the process of reason. As such, I think a person who made a very concerted effort could be completely logical - could apply logic to the totality of their conscious thoughts. Root assumptions need not be proved irrefutably - they need only be examined rationally and found to be worthwhile, whether by evidence in their favour or probable evidence in their favour. What is worthwhile? Well, therein lies the circularity again: in the end, there is an innate human judgement that must take place as to what reason is, the same judgement that invented logic in the first place. Is circularity invalid or a cause for slandering the fair name of all logic? I don't think so - we are only circular in our thinking to the extent that the fundamental question of being is circular: "Why does anything exist at all? Why not just nothingness?" Our existence is centered around the fact that the universe does in fact exist - that is the root assumption of everything.

A logical system can be thought as a set of foundative axioms (which are statements), and a set of rules to deduce a new statement from already existing ones.
Actually, there could be different sets of rules for the same set of axioms, generating different logics.

Anyway, keeping this in mind it could be quite difficult to find out what are the implicit axioms of a man reasoning system, because with don't know neither the deducing rules.